Let \(X, Y\)  be Hilbert spaces and let \(K: X \longrightarrow Y\)  be a bounded
linear operator with \(\mathcal {D}(K) = X\) . Then we recall that the dual operator
\(K^{\prime }\)  is the maximum operator among operators \(\widehat{K}: Y^{\prime } \longrightarrow X^{\prime }\)  with
\(\mathcal {D}(\widehat{K}) \subset Y^{\prime }\)  such that
\(_{X^{\prime }}{<} \widehat{K}y, x>_{X} \,=\, {_{Y^{\prime }}{<} y, Kx>_{Y}}\) 
for each \(x\in X\)  and
\(y \in \mathcal {D}(\widehat{K}) \subset Y^{\prime }\)  (e.g., [1]}).
